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MATH408: Probability & Statistics Summer 1999 WEEK 5. Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu

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math408 probability statistics summer 1999 week 5
MATH408: Probability & StatisticsSummer 1999WEEK 5

Dr. Srinivas R. Chakravarthy

Professor of Mathematics and Statistics

Kettering University

(GMI Engineering & Management Institute)

Flint, MI 48504-4898

Phone: 810.762.7906

Email: schakrav@kettering.edu

Homepage: www.kettering.edu/~schakrav

joint pdf
Joint PDF
  • So far we saw one random variable at a time. However, in practice, we often see situations where more than one variable at a time need to be studied.
  • For example, tensile strength (X) and diameter(Y) of a beam are of interest.
  • Diameter (X) and thickness(Y) of an injection-molded disk are of interest.
joint pdf cont d x and y are continuous
Joint PDF (Cont’d)X and Y are continuous
  • f(x,y) dx dy = P( x < X < x+dx, y < Y < y+dy) is the probability that the random variables X will take values in (x, x+dx) and Y will take values in (y,y+dy).
  • f(x,y)  0 for all x and y and
independence
Independence

We say that two random variables X and Y are independent if and only if

P(XA, YB) = P(XA)P(YB) for all A and B.

groundwork for inferential statistics
Groundwork for Inferential Statistics
  • Recall that, our primary concern is to make inference about the population under study.
  • Since we cannot study the entire population we rely on a subset of the population, called sample, to make inference.
  • We saw how to take samples.
  • Having taken the sample, how do we make inference on the population?
slide18

Figure 3-36 (b) Probability density function of the average of

8 pull-off force measurements in Example 3-33.

slide19

Figure 3-36 (c) Probability density Probability density function

function of the sample variance of 8 pull-off force

measurements in Example 3-33.

central limit theorem
Central Limit Theorem
  • One of the most celebrated results in Probability and Statistics
  • History of CLT is fascinating and should read “The Life and Times of the Central Limit Theorem” by William J. Adams
  • Has found applications in many areas of science and engineering.
clt cont d
CLT (cont’d)
  • A great many random phenomena that arise in physical situations result from the combined actions of many individual ones.
  • Shot noise from electrons; holes in a vacuum tube or transistor; atmospheric noise, turbulence in a medium, thermal agitation of electrons in a conductor, ocean waves, fluctuations in stock market, etc.
clt cont d1
CLT (cont’d)
  • Historically, the CLT was born out of the investigations of the theory of errors involved in measurements, mainly in astronomy.
  • Abraham de Moivre (1667-1754) obtained the first version.
  • Gauss, in the context of fitting curves, developed the method of Least Squares, which lead to normal distribution.
homework problems
HOMEWORK PROBLEMS

Sections 3.11 through 3.12

109,111, 114-116-119, 121-123, 129-130

slide35

Tests of Hypotheses

  • Two types of hypotheses: Null (H0)and alternative (H1)
basic ideas in tests of hypotheses
Basic Ideas in Tests of Hypotheses
  • Set up H0 and H1. For a one-sided case, make sure these are set correctly. Usually these are done such that type 1 error becomes “costly” error.
  • Choose appropriate test statistic. This is usually based on the UMV estimator of the parameter under study.
  • Set up the decision rule if  = P(type 1 error) is specified. If not, report a p-value.
  • Choose a random sample and make the decision.
setting up h o and h 1
Setting up Hoand H1
  • Suppose that the manufacturer of airbags for automobiles claims that the mean time to inflate airbag is no more than 0.1 second.
  • Suppose that the “costly error” is to conclude erroneously that the mean time is < 0.1.
  • How do we set up the hypotheses?
test on using normal
Test on µ using normal
  • Sample size is large
  • Sample size is small, population is approximately normal with known .
slide41

DNR Region

µ

CP_1

CP_2

example page 142
Example (page 142)
  • µ = Mean propellant burning rate (in cm/s).
  • H0:µ = 50 vs H1:µ  50.
  • Two-sided hypotheses.
  • A sample of n=10 observations is used to test the hypotheses.
  • Suppose that we are given the decision rule.
  • Question 1: Compute P(type 1 error)
  • Question 2: Compute P(type 2 error when µ =52.
confidence interval
Confidence Interval
  • Recall point estimate for the parameter under study.
  • For example, suppose that µ= mean tensile strength of a piece of wire.
  • If a random sample of size 36 yielded a mean of 242.4psi.
  • Can we attach any confidence to this value?
  • Answer: No! What do we do?
confidence interval cont d
Confidence Interval (cont’d)
  • Given a parameter, say, , let denote its UMV estimator.
  • Given , 100(1-  )% CI for is constructed using the sampling (probability) distribution of as follows.
  • Find L and U such that P(L < < U) = 1- .
  • Note that L and U are functions of .